Proofs - Schr¨ odinger Equation Sudarshan Balakrishnan Lemma 2 The Fourier coefficients of a function f x are q-periodic in their indices so c k q

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Unformatted text preview: Schr¨ odinger Equation Sudarshan Balakrishnan June 14, 2011 Lemma 2. The Fourier coefficients of a function f ( x ) are q-periodic in their indices, so c k + q = c k for all k , if and only if the series represents a linear combination of q (periodically extended) delta functions concentrated at the rational nodes x j = 2 πj/q for j = 0 , 1 ,...,q- 1 : f ( x ) = q- 1 X j =0 a j δ ( x- 2 πj/q ) (1) Proof ←- : Here, we have that the series represents a linear combination of q delta functions concentrated at the rational nodes x j = 2 πj/q for j = 0 , 1 ,...,q- 1. Now, to justify the integrability of the delta function, let us construct our delta function as follows: δ ( x ) = lim → 1 2 e- x 2 . (2) Now, for points x j = 2 πj/q : δ ( x- 2 πj/q ) = lim → 1 2 e- ( x- 2 πj/q ) 2 . (3) Constructing the fourier coefficients similar to Theorem1 we have that, c j,q k = 1 2 π Z 2 π δ ( x- 2 πj/q ) e- ikx dx. (4) We will simplify (4), using the following lemma: 1 Lemma: For a given dirac delta function δ ( x ) and a continuous function f ( x ) at x : f ( x ) = Z + ∞-∞ δ ( x- x ) f ( x ) dx. (5) Proof: First, consider an...
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This note was uploaded on 11/19/2011 for the course MATH 101 taught by Professor Wormer during the Spring '08 term at UCSD.

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Proofs - Schr¨ odinger Equation Sudarshan Balakrishnan Lemma 2 The Fourier coefficients of a function f x are q-periodic in their indices so c k q

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